We obtain new and simple characterizations for the boundedness and compactness of weighted composition operators on the Fock space over C. We also describe all weighted composition operators that are normal or isometric.

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a b s t r a c tWe study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form (1 − ⟨z, w⟩) −γ for γ > 0. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.

Abstract. We obtain simple characterizations of unilateral and bilateral weighted shift operators that are m -isometric. We show that any such operator is a Hadamard product of 2 -isometries and 3 -isometries. We also study weighted shift operators whose powers are m -isometric.Mathematics subject classification (2010): 47B37, 47A65.